Hmmm, I'm thinking maybe it's the solenoid.
It doesn't have a solenoid. A solenoid is used in later starters. When you mash the foot starter the first thing that happens is the gear in the starter engages the gear on the flywheel - that's what the solenoid does in newer starters.
If you hotwire your starter without doing anything else it should spin freely, if not it's defective. If it is defective pull it out, put it on the bench, turn the gear a little and hotwire it again. If it still doesn't turn, you got problems. If it does spin you probably have a "dead spot". Either way, a rebuild is in order.
Just hope you didn't burn it up trying to get your engine started. Not likely, but possible if you ran it for long periods of time with a 12v battery.
